14.127 Behavioral Economics
(Lecture 1)
Xavier Gabaix
February 5, 2003
1 Overview
• Instructor: Xavier Gabaix
• Time 4-6:45/7pm, with 10 minute break.
• Requirements: 3 problem sets and
Term paper due September 15, 2004 (meet Xavier in May
to talk
about it)
2 Some Psychology of Decision Making
2.1 Prospect Theory (Kahneman-Tversky, Econometrica
79)
Consider gambles with two outcomes: x with probability p,
and y with probability 1-p where x ≥ 0 ≥ y .
• Expected utility (EU) theory says that if you start with
wealth W then the (EU) value of the gamble is
• Prospect theory (PT) says that the (PT) value of the game
is
where π is a probability weighing function. In standard
theory π is linear.
• In prospect theory π is concave first and then convex, e.g.
• for some β∈ (0 , 1). The figure gives π ( p ) for β = .8
2.1.1 What does the introduction of the weighing function π
mean?
• π(p) ＞ p for small p. Small probabilities are overweighted, too
salient.
E.g. people play a lottery. Empirically, poor people and less
educated
people are more likely to play lottery. Extreme risk aversion.
• π(p) ＜ p for p close to 1. Large probabilities are underweight.
In applications in economics π(p) = p is often used except for
lotteries and insurance
2.1.2 Utility function u
• We assume that u ( x ) is increasing in x, convex for loses,
concave for gains, and first order concave at 0 that is
• A useful parametrization
• The graph of u ( x ) for λ = 2 and β = .8 is given below
2.1.3 Meaning - Fourfold pattern of risk aversion u
• Risk aversion in the domain of likely gains
• Risk seeking in the domain of unlikely gains
• Risk seeking in the domain of likely losses
• Risk aversion in the domain of unlikely losses
2.1.4 How robust are the results?
• Very robust: loss aversion at the reference point, λ>1
• Robust: convexity of u for x < 0
• Slightly robust: underweighting and overweighting of
probabilities π ( p ) ≷ p
2.1.5 In applications we often use a simplified PT
(prospect theory)
and
2.1.6 Second order risk aversion of EU
• Consider a gamble x +σ and x -σ with 50 : 50 chances.
• Question: what risk premium πwould people pay to avoid the
small risk σ ?
• We will show that as σ → 0 this premium is O (σ2 ). This is
called second order risk aversion.
• In fact we will show that for twice continuously differentiable
utilities:
where ρ is the curvature of u at 0 that is
• The risk premium π makes the agent with utility function u
indifferent between
• Assume that u is twice differentiable and take a look at the
Taylor expansion of the above equality for small σ. .
or
• Since π(σ ) is much smaller than σ , so the claimed
approximation is true. Formally, conjecture the
approximation, verify it, and use
the implicit function theorem to obtain uniqueness of the
function π defined implicitly be the above approximate
equation.
2.1.7 First order risk aversion of PT
• Consider same gamble as for EU.
• We will show that in PT, as σ→0, the risk premium π is of
the order of σ when reference wealth x = 0. This is called
the first order risk aversion.
• Let’s compute π for u (x) = xα for x ≥ 0 and u (x) =－λ|x|α
for
x ≤ 0.
• The premium π at x = 0 satisfies
or
where k is defined appropriately.
2.1.8 Calibration 1
• Take
CES, utility
, i.e. a constant elasticity of substitution,
• Gamble 1
$50,000 with probability 1/2
$100,000 with probability 1/2
• Gamble 2. $ x for sure.
• Typical x that makes people indifferent belongs to (60k, 75k)
(though
some people are risk loving and ask for higher x.
• Note the relation between x and the elasticity of
substitution γ:
x 70k 63k58k 54k 51.9k 51.2k
γ 1
3
5
10
20
30
Right γ seems to be between 1 and 10.
• Evidence on financial markets calls for γ bigger than 10.
This is the equity premium puzzle.
2.1.9 Calibration 2
• Gamble 1
$10.5 with probability ½
$-10 with probability ½
• Gamble 2. Get $0 for sure.
• If someone prefers Gamble 2, she or he satisfies
Here, π = $.5 and σ = $10.25. We know that in EU
And thus with CES utility
forces large γ as the wealth W is larger than 105
easily.
2.1.10 Calibration Conclusions
• In PT we have π* = kσ. For γ = 2, and σ= $.25 the risk
premium
is π* = kσ= $.5 while in EU π* = $.001.
• If we want to fit an EU parameter γto a PT agent we get
and this explodes as σ → 0.
• If someone is averse to 50-50 lose $100/gain g for all wealth
levels
then he or she will turn down 50-50 lose L /gain G in the table
L\g
$101
$105
$110
$125
$400
$400
$420
$550
$1,250
$800
$800
$1,050
$2,090
∞
$1000
$1,010
$1,570
∞
∞
$2000
$2,320
∞
∞
∞
$10,000
∞
∞
∞
∞
2.2 What does it mean?
• EU is still good for modelling.
• Even behavioral economist stick to it when they are not
interested in risk taking behavior, but in fairness for
example.
• The reason is that EU is nice, simple, and parsimonious.
2.2.1 Two extensions of PT
• Both outcomes, x and y , are positive, 0 < x < y .Then,
Why not
? Because it
becomes self-contradictory when x = y and we stick to K-T
calibration that putsπ(.5) < .5 .
• Continuous gambles, distribution f (x)
EU gives:
PT gives: